Why is the camera going inside my mesh, when the camera is not inside?
The answer to this problem is simple.  Blender has a feature where you can select an object, hit ctrl-0, and that object becomes the "camera."  Thus, the camera view was inside my object.  This is pretty neat!  I'm not sure how to use it in a beneficial way, but it must be good for something.
If you want to set the camera view back to your original camera, select your camera and hit ctrl-0.